Simplify and expand the following expression: $ \dfrac{2}{x + 9}- \dfrac{4}{2x - 16}+ \dfrac{5x}{x^2 + x - 72} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{4}{2x - 16} = \dfrac{4}{2(x - 8)}$ We can factor the quadratic in the third term: $ \dfrac{5x}{x^2 + x - 72} = \dfrac{5x}{(x + 9)(x - 8)}$ Now we have: $ \dfrac{2}{x + 9}- \dfrac{4}{2(x - 8)}+ \dfrac{5x}{(x + 9)(x - 8)} $ The least common multiple of the denominators is: $ (x + 9)(x - 8)$ In order to get the first term over $(x + 9)(x - 8)$ , multiply by $\dfrac{2(x - 8)}{2(x - 8)}$ $ \dfrac{2}{x + 9} \times \dfrac{2(x - 8)}{2(x - 8)} = \dfrac{4(x - 8)}{(x + 9)(x - 8)} $ In order to get the second term over $(x + 9)(x - 8)$ , multiply by $\dfrac{x + 9}{x + 9}$ $ \dfrac{4}{2(x - 8)} \times \dfrac{x + 9}{x + 9} = \dfrac{4(x + 9)}{(x + 9)(x - 8)} $ In order to get the third term over $(x + 9)(x - 8)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{5x}{(x + 9)(x - 8)} \times \dfrac{2}{2} = \dfrac{10x}{(x + 9)(x - 8)} $ Now we have: $ \dfrac{4(x - 8)}{(x + 9)(x - 8)} - \dfrac{4(x + 9)}{(x + 9)(x - 8)} + \dfrac{10x}{(x + 9)(x - 8)} $ $ = \dfrac{ 4(x - 8) - 4(x + 9) + 10x} {(x + 9)(x - 8)} $ Expand: $ = \dfrac{4x - 32 - 4x - 36 + 10x}{2x^2 + 2x - 144} $ $ = \dfrac{-68 + 10x}{2x^2 + 2x - 144}$ Simplify: $ = \dfrac{-34 + 5x}{x^2 + x - 72}$